a ship traveling at a speed of 48.0 mph at a compass heading of 145°. the current is flowing due south at a speed of 9mph. find the resultant speed of the ship due to the current and the resultant compass heading of the ship
To find the resultant speed of the ship due to the current, we can consider the ship's velocity vector and the current's velocity vector as vectors that can be added together.
First, we need to convert the speed of the ship and the current from mph to a more convenient unit, such as knots. Since 1 knot is equal to 1 nautical mile per hour, we have:
Ship's speed in knots = 48.0 mph * 0.868976 knots/mph = 41.731488 knots
Current's speed in knots = 9 mph * 0.868976 knots/mph = 7.820784 knots
Next, let's determine the x and y components of the ship's velocity vector. The x component represents the east-west direction, and the y component represents the north-south direction. We'll use trigonometry to calculate these components.
x component = ship's speed * cos(compass heading)
y component = ship's speed * sin(compass heading)
where the compass heading is measured clockwise from true north.
x component = 41.731488 knots * cos(145°) = -18.878144 knots
y component = 41.731488 knots * sin(145°) = 37.147113 knots
Now, let's determine the x and y components of the current's velocity vector. Since the current is flowing due south, it only has a y component.
x component of current = 0 knots
y component of current = -7.820784 knots
To find the resultant x and y components, we add the corresponding components of the ship's velocity and the current's velocity:
resultant x component = x component of ship's velocity + x component of current's velocity
resultant y component = y component of ship's velocity + y component of current's velocity
resultant x component = -18.878144 knots + 0 knots = -18.878144 knots
resultant y component = 37.147113 knots - 7.820784 knots = 29.326329 knots
Now, we can calculate the resultant speed of the ship using the Pythagorean theorem:
resultant speed = sqrt((resultant x component)^2 + (resultant y component)^2)
resultant speed = sqrt((-18.878144 knots)^2 + (29.326329 knots)^2) = 35.15 knots (rounded to two decimal places)
Finally, to find the resultant compass heading of the ship, we need to use inverse trigonometric functions:
resultant compass heading = atan2(resultant y component, resultant x component)
resultant compass heading = atan2(29.326329 knots, -18.878144 knots) = 123.60° (rounded to two decimal places)
Therefore, the resultant speed of the ship due to the current is approximately 35.15 knots, and the resultant compass heading of the ship is approximately 123.60°.